ON THE INTEGRABILITY OF SYMPLECTIC ` MONGE-AMPERE EQUATIONS Eugene Ferapontov Department of Mathematical Sciences Loughborough University
[email protected] Collaboration: B Doubrov, L Hadjikos, K Khusnutdinova Current Problems in Differential Calculus over Commutative Algebras, Secondary Calculus, and Solution Singularities of Nonlinear PDEs 13-16 June 2011, Vietri Sul Mare, Italy
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Plan: • Dispersionless Hirota type equations ` equations • Symplectic Monge-Ampere
• Integrability: the method of hydrodynamic reductions ` equations in 3D • Symplectic Monge-Ampere ` equations in 4D. Classification. Geometry • Symplectic Monge-Ampere
• Generalisations References E.V. Ferapontov, L. Hadjikos and K.R. Khusnutdinova, Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, International Mathematics Research Notices, (2010) 496-535; arXiv: 0705.1774 (2007). ` equations, B. Doubrov and E.V. Ferapontov, On the integrability of symplectic Monge-Ampere Journal of Geometry and Physics, 60 (2010) 1604-1616. 2
Dispersionless Hirota type equations Three dimensions, u
= u(x, y, t): F (uxx , uxy , uyy , uxt , uyt , utt ) = 0
Examples (dispersionless limits of 2+1D integrable hierarchies):
1 utx − u2xx = uyy 2 uxx + uyy = eutt
dKP equation Boyer − Finley equation Higher dimensions, u
= u(x1 , ..., xn ): F (uij ) = 0
Solutions can be interpreted as Lagrangian submanifolds in the symplectic space with coordinates x1 , ..., xn , u1 , ..., un whose Gaussian image belongs to the hypersurface in the Lagrangian Grassmannian specified by F (uij ) Integrability? Classification? Geometry? 3
= 0.
` equations Symplectic Monge-Ampere = (uij ) be the Hessian matrix of a function u(x1 , ..., xn ). Symplectic ` equations are linear combinations of all possible minors of U Monge-Ampere
Let U
Examples:
First heavenly equation
u13 u24 − u14 u23 = 1
Second heavenly equation
u13 + u24 + u11 u22 − u212 = 0
Husain equation
u11 + u22 + u13 u24 − u14 u23 = 0
6D heavenly equation
u15 + u26 + u13 u24 − u14 u23 = 0
Special Lagrangian 3 − folds
Hess u = 4u
Affine spheres
Hess u = 1
Equivalence group Sp(2n) acts by linear transformations of x1 , ..., xn , u1 , ..., un Integrability? Classification? Geometry? 4
Special Lagrangian 3-folds Consider the space C3 with coordinates z 1 , z 2 , z 3 Symplectic form
(z k = xk + iuk )
ω = du1 ∧ dx1 + du2 ∧ dx2 + du3 ∧ dx3
Holomorphic volume form
Ω = dz 1 ∧ dz 2 ∧ dz 3
Im Ω = −du1 ∧du2 ∧du3 +du1 ∧dx2 ∧dx3 +dx1 ∧du2 ∧dx3 +dx1 ∧dx2 ∧du3 Special Lagrangian 3-folds are specified by the equations ω
= Im Ω = 0
In general: symplectic space R2n with coordinates x1 , ..., xn , u1 , ..., un Symplectic form
ω = du1 ∧ dx1 + ... + dun ∧ dxn
Constant coefficient differential n-form Φ in dxk , duk ` equations are specified by the equations ω Symplectic Monge-Ampere Manifestly Sp(2n) invariant
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=Φ=0
Integrability: the method of hydrodynamic reductions Applies to quasilinear systems
A(u)ux + B(u)uy + C(u)ut = 0 Consists of seeking N-phase solutions
u = u(R1 , ..., RN ) The phases Ri (x, y, t) are required to satisfy a pair of commuting equations
Ryi = µi (R)Rxi , ∂j µi Commutativity conditions: µj −µi
=
Rti = λi (R)Rxi
∂j λ i λj −λi
Definition A quasilinear system is said to be integrable if, for any number of phases N, it possesses infinitely many hydrodynamic reductions parametrised by N arbitrary functions of one variable. 6
Example of dKP 1 2 uxt − uxx = uyy 2 Quasilinear form: set v = uxx , w = uxy vt − vvx = wy , vy = wx N -phase solutions: v = v(R1 , ..., RN ), w = w(R1 , ..., RN ) where Ryi = µi (R)Rxi ,
Rti = λi (R)Rxi
Then
∂i w = µi ∂i v,
λi = v + (µi )2
Equations for v(R) and µi (R) (Gibbons-Tsarev system):
∂j v , ∂j µ = j µ − µi i
∂i v∂j v ∂i ∂j v = 2 j (µ − µi )2
In involution! General solution depends on N arbitrary functions of one variable. 7
Generalised dKP uxt − f (uxx ) = uyy Quasilinear form: set v
= uxx , w = uxy vt − f (v)vx = wy , vy = wx
N -phase solutions: v = v(R1 , ..., RN ), w = w(R1 , ..., RN ) where Ryi = µi (R)Rxi ,
Rti = λi (R)Rxi
Then
∂i w = µi ∂i v,
λi = f 0 (v) + (µi )2
Generalised Gibbons-Tsarev system:
∂j µi = f 00 (v) Involutivity
∂j v , j i µ −µ
∂i ∂j v = 2f 00 (v)
←→ f 000 = 0 8
∂i v∂j v (µj − µi )2
General case utt = f (uxx , uxy , uyy , uxt , uyt ) The integrability conditions reduce to a system of third order PDEs for f :
d3 f = S(f, df, d2 f ) In involution!
Theorem The moduli space of integrable equations of the dispersionless Hirota type is
21-dimensional. The action of the equivalence group Sp(6) on the moduli space of integrable equations possesses an open orbit
• Parametrisation by generalised hypergeometric functions (Odesskii-Sokolov) • Geometric characterisation of the associated GL(2, R) structures (Smith) 9
Further examples uxy 1 utt = + η(uxx )u2xt , uxt 6 η solves the Chazy equation η 000 + 2ηη 00 = 3(η 0 )2 . uxy + uxt uyt r(utt ) = 0, r solves the third order ODE 2
r000 (r0 − r2 ) − r002 + 4r3 r00 + 2r03 − 6r2 r0 = 0. General solution
∞ X (−1)n nq n t , q = e r(t) = 1 − 8 n 1 − q n=1
r(t) is the Eisenstein series associated with the congruence subgroup Γ0 (2) of the modular group. 10
Geometric picture
Equation F
= 0 −→ Hypersurface M 5 ⊂ Λ6
Solutions −→ Lagrangian manifolds whose Gaussian images belong to M 5
Sp(6) −→ Equivalence group of the problem Classification of equations of the dispersionless Hirota type up to Sp(6) equivalence is equivalent to the study of Sp(6) geometry of hypersurfaces
M 5 ⊂ Λ6 11
Geometry of the Lagrangian Grassmannian Action of Sp(6) on the Lagrangian Grassmannian Λ6 :
A
B
C
D
˜ = (AU + B)(CU + D)−1 ∈ Sp(6) =⇒ U ˜ ) = (...)det dU det (dU
Theorem 3 The group of conformal automorphisms of the symmetric cubic form det dU is isomorphic to Sp(6) Objects in P 5
= P T Λ6 : cubic hypersurface det dU = 0
Its singular locus is the Veronese surface V 2
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⊂ P5
Geometry of a hypersurface M 5
⊂ Λ6
The intersection of T M 5 with the Veronese surface V 2 specifies in T M 5 a rational normal curve of degree four, that is, a curve equivalent to
(1 : t : t2 : t3 : t4 ) ⊂ P 4
Thus, M 5 is supplied with a Gl(2, R)-structure
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Geometric interpretation of the integrability
1-component hydrodynamic reductions −→ secant curves 2-component hydrodynamic reductions −→ bisecant surfaces 3-component hydrodynamic reductions −→ trisecant 3-folds
Integrability ←→ existence of sufficiently many trisecant 3-folds
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` equations in 2D Symplectic Monge-Ampere Hessian matrix
U =
u11 u12
u12 u22
` equations are of the form Symplectic Monge-Ampere
M2 + M1 + M0 = 0 Explicitly,
(u11 u22 − u212 ) + au11 + bu12 + cu22 + d = 0 Any such equation is linearisable by a transformation from Sp(4)
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Geometry in 2D Symplectic space with coordinates x1 , x2 , u1 , u2 . Lagrangian planes form the Lagrangian Grassmannian Λ3
u1 u2
=
Plucker embedding of Λ3 in P 4 is (1 ¨
u11 u12
u12 u22
x1
x2
: u11 : u12 : u22 : u11 u22 − u212 )
` equations ←→ hyperplanes in P 4 Symplectic Monge-Ampere 16
` equations in 3D Symplectic Monge-Ampere Hessian matrix
u11 U = u12 u13
u13
u22
u23
u23
u33
u12
` equations are of the form Symplectic Monge-Ampere
M3 + M2 + M1 + M0 = 0 In 3D there exist three essentially different canonical forms modulo the equivalence group Sp(6) (Lychagin, Rubtsov, Chekalov, Banos):
u11 = u22 + u33 ,
Hess u = 4u,
Hess u = 1
Integrability ←→ linearisability (not true in dim 4 and higher: the first heavenly equation, u13 u24 − u14 u23 = 1, is integrable, but not linearisable). 17
Geometry in 3D Symplectic space with coordinates x1 , x2 , x3 , u1 , u2 , u3 . Lagrangian planes form the Lagrangian Grassmannian Λ6
Plucker embedding of Λ6 in P 13 ¨
` equations in 3D ←→ Integrable symplectic Monge-Ampere hyperplanes tangential to Λ6 ⊂ P 13 18
` equations in 4D Symplectic Monge-Ampere ` equation in 4D for u(x1 , x2 , x3 , x4 ) and Consider a symplectic Monge-Ampere take its traveling wave reduction to 3D,
u = u(x1 + αx4 , x2 + βx4 , x3 + γx4 ) + Q(x, x), where Q(x, x) is an arbitrary quadratic form.
Integrability in 4D ←→ linearisability of all traveling wave reductions to 3D In particular, all traveling wave reductions of the first heavenly equation,
u13 u24 − u14 u23 = 1, are linearisable.
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Classification of integrable equations in 4D linear wave
u11 − u22 − u33 − u44 = 0
first heavenly
u13 u24 − u14 u23 = 1
second heavenly
u13 + u24 + u11 u22 − u212 = 0
modified heavenly
u13 = u12 u44 − u14 u24
Husain
u11 + u22 + u13 u24 − u14 u23 = 0
general heavenly
αu12 u34 + βu13 u24 + γu14 u23 = 0, α + β + γ = 0
Conjecture In dimensions D ≥ 4, any integrable equation of the form ` type F (uij ) = 0 is necessarily of the symplectic Monge-Ampere Not true in 3D: take the dKP equation uxt
− 12 u2xx = uyy
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Geometry in 4D Symplectic space with coordinates x1 , x2 , x3 , x4 , u1 , u2 , u3 , u4 . Lagrangian planes form the Lagrangian Grassmannian Λ10
Plucker embedding of Λ10 in P 41 is covered by a 7-parameter family of Λ6 ¨
` equations ←→ hyperplanes Integrable symplectic Monge-Ampere tangential to Λ10 ⊂ P 41 along a four-dimensional subvariety X 4 which meets all Λ6 ⊂ P 41 21
Generalisations Gr(d, n): Grassmannian of d-dimensional subspaces of a linear space V n . X ⊂ Gr(d, n): submanifold of the Grassmannian, dim X = n − d. Σ(X): differential system governing d-dimensional submanifolds of V n whose Gaussian image is contained in X . Conjectures
• For d ≥ 3, n − d ≥ 2 the moduli space of integrable systems Σ(X) is finite dimensional
• For d ≥ 4, n − d ≥ 2 any integrable system Σ(X) must be linearly ` property: systems of degenerate (generalisation of the Monge-Ampere ` type correspond to linear sections of the Plucker Monge-Ampere ¨ embedding of Gr(d, n)).
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